Using this algorithm, we reproduce the
well known chamber structure for DT invariants along the large volume slice
 [1]}, [2]}, [3]},
consisting of a finite nested sequence of walls of marginal stability,
such that
the index
vanishes inside the innermost wall and is equal to the index \(\Omega _\infty (\infty )\)  counting Gieseker-semistable sheaves outside the outermost wall.
For illustration, in §REF  and §REF  we determine the trees contributing to \(\Omega _{\infty }(\gamma )\) 
for \(\gamma =[1,0,1-n]\)  and \(\gamma =[0,d,\textnormal {ch}_2]\)  for low values of \(n\)  and \(d\) .
In the first case, the moduli space of Gieseker semi-stable sheaves coincides with the
Hilbert scheme of \(n\)  points on \({P}^2\) , the index of which is well-known
[4]} (higher rank examples are considered in Appendix §).
In the second case
\(r=0\) , we recover the genus zero Gopakumar-Vafa invariants \(N_d^{(0)}\)  in the unrefined limitMore generally, the refined Gieseker index computes the character \(\sum _{j_L,j_R} N^{(j_L, j_R)}_d \chi _{j_L}(y_L) \chi _{j_R}(y_R)\)  on the diagonal \(y_L=y_R=y\) , where \(N^{(j_L, j_R)}_d\)  are the refined BPS invariants [5]}, [6]}. It is an interesting open question
to generalize the scattering diagram away from the Nekrasov-Shatashvilii limit \(y_L=y_R\) . \(y\rightarrow 1\) .
We further match the contributing trees with the known stratification of the moduli space of Gieseker stable sheaves, extending the observations in [7]}.
